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Find the equation of the plane parallel to two lines

  1. Sep 6, 2016 #1
    1. The problem statement, all variables and given/known data
    Let A, B and C be distinct vectors in V3 with B and C non-parallel.
    a. Find an equation for the plane containing both the line through A parallel to B and the line through A parallel to C.
    b. Verify that the two lines actually lie in the plane.

    2. Relevant equations


    3. The attempt at a solution
    I can't see any situation where a plane containing A can be parallel to a plane containing C if they are distinct vectors (distinct as in non-parallel)
     
  2. jcsd
  3. Sep 6, 2016 #2

    Mark44

    Staff: Mentor

    What is V3? Presumably it's a 3-dimensional vector space. If so, is it different from ##\mathbb{R}^3##?
    This is a very strange description. The context here seems to be that A is a point, but the earlier description is that A is a vector. I would guess that what they're calling vector A is the vector ##\vec{OA}##, with O being the origin and A being the endpoint of the vector. Otherwise this problem doesn't make much sense.

    Was the problem statement originally in some other language?
     
  4. Sep 6, 2016 #3

    andrewkirk

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    The question is not well expressed. Here is what I think they mean.

    Let A,B,C be three points in ##\mathbb R^3## and denote the origin by ##O##.

    Let L1 be a line through A that is parallel to line segment ##\bar{OB}##.
    Let L2 be a line through A that is parallel to line segment ##\bar{OC}##.

    (b) Show that there is a plane that contains both L1 and L2; and
    (a) find the equation of that plane.
     
  5. Sep 6, 2016 #3

    Ray Vickson

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    While the question is not absolutely clear, I think one interpretation of it is that we have two vectors ##\vec{B}## and ##\vec{C}## in a 3-dimensional space; these are, perhaps, like force vectors or velocity vectors, having tails and heads not necessarily at the origin. Assuming that is the case, make a copy ##\vec{C}'## of ##\vec{C}## but whose tail coincides with the tail of ##\vec{B}##, so that ##\vec{B}## and ##\vec{C}'## emanate from the same point in space, but have different directions. There certainly IS a plane P that contains both vectors ##\vec{B}## and ##\vec{C}'## (that is, which contains the three points that lie at the two ends of these vectors). Any Plane P that is parallel to P will be parallel to both vectors ##\vec{B}## and ##\vec{C}## (the original ##\vec{C}##), and now all you need to is figure out which such plane passes through the point A which lies at the end of the vector ##\vec{A}## (assuming that this last vector has its tail at the origin).
     
  6. Sep 7, 2016 #4
    I copied the problem straight off the problem set; I'll try to clarify with the teacher today
     
  7. Sep 7, 2016 #5
    There's no difference, my teacher just likes to use V3 as a distinction from R3 (for whatever reason) for the first few days of the class
     
  8. Sep 7, 2016 #6
    OK so apparently, we are supposed to find a plane that has both B and C in it, and A intersecting it. It makes sense because a plane can be parallel to many vectors.
    The equation of such a plane can be [itex]\left( \left< x,\quad y,\quad z \right> -\overrightarrow { OA } \right) \cdot \overrightarrow { B } \times \overrightarrow { C } =0[/itex] right?
    Then I guess I just do part b by plugging in OA + tB and OA + tC.
     
    Last edited: Sep 7, 2016
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